# What does min max in optimization problems mean?

With $$A$$ and $$B$$ being different decision variables, does $$\min_{A} \max_{B}$$ in front of an objective function specify a specific order of optimization? In other words, is it saying "minimize $$A$$ while maximizing $$B$$", or "minimize $$A$$ while holding the maximized $$B$$ fixed", or something else?

What if the places are swapped, or some other combination?

$$\max_B \min_A$$

Finally, can the min-max prefix appear for any sort of optimization problem, i.e. quadratic programming (convex optimization), linear programming, dynamic programming? How does it differ from "minimax"?

The notation $$\min\limits_A \max\limits_B f(A,B)$$ does not mean to minimize $$A$$ or maximize $$B$$. It means that, for each fixed value of $$A$$, you find a $$B$$ value that maximizes $$f(A,B)$$, and you find a value of $$A$$ that minimizes that maximum value. If it helps, you can think of the "inner problem" as $$g(A) = \max\limits_B f(A,B)$$, and then the "outer" problem is $$\min\limits_A g(A)$$. It is also called a minimax problem. Yes, you can interchange the roles to obtain a maximin problem. More generally, you can have an arbitrary combination of any number of $$\min$$ and $$\max$$ operators.